Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ₑᵉ^³ dx / (x ln x ln²(ln x))
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.21
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.21Chapter 7, Problem 7.1.21
7–28. Derivatives Evaluate the following derivatives.
d/dx (e^{-10x²})
Verified step by step guidance1
Step 1: Recognize that the function involves the exponential function e raised to a power, specifically e^{-10x²}. The derivative of e^u with respect to x is e^u * du/dx, where u is a function of x.
Step 2: Identify the inner function u = -10x². To apply the chain rule, we need to compute the derivative of u with respect to x.
Step 3: Differentiate u = -10x² with respect to x. Using the power rule, the derivative of x² is 2x, and multiplying by the constant -10 gives du/dx = -20x.
Step 4: Apply the chain rule. The derivative of e^{-10x²} with respect to x is e^{-10x²} * (-20x), combining the derivative of the outer function e^u and the inner function u = -10x².
Step 5: Simplify the expression. The derivative is -20x * e^{-10x²}. This is the simplified form of the derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. Understanding how to compute derivatives is essential for analyzing the behavior of functions, including their maxima, minima, and points of inflection.
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Derivatives
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay, depending on the sign of 'b'. In the context of derivatives, the derivative of an exponential function is proportional to the function itself, which simplifies the differentiation process.
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions that involve exponentials, as seen in the differentiation of e^{-10x²}.
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Related Practice
Textbook Question
101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y
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Textbook Question
Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.
Textbook Question
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁ᵉ^² dx/x√(ln²x + 1)
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Textbook Question
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Textbook Question
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